To substitute the slope \( m \) into the linear equation \( y = mx + b \), we first identify the values we have. From your provided points, we can find the slope:
The two points are:
- \( (4, 120) \)
- \( (7, 150) \)
The slope \( m \) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the values from the points: \[ m = \frac{150 - 120}{7 - 4} = \frac{30}{3} = 10 \]
You mentioned that the slope is \( 10 \). We can now substitute \( m \) into the equation \( y = mx + b \).
So, we have: \[ y = 10x + b \]
Next, we need to determine the \( y \)-intercept \( b \). To do this, we can use one of the points provided (let’s use \( (4, 120) \)) to find \( b \): \[ 120 = 10(4) + b \] \[ 120 = 40 + b \] \[ b = 120 - 40 = 80 \]
Now we can write the complete linear equation: \[ y = 10x + 80 \]
Thus, the height of the tree in inches can be represented by the equation: \[ y = 10x + 80 \]
In this equation:
- \( y \) is the height of the tree in inches,
- \( x \) is the number of years since the base year (the starting point),
- The slope \( 10 \) indicates that the tree grows \( 10 \) inches for each year that passes,
- The \( y \)-intercept \( 80 \) indicates that at year \( 0 \) (the base year), the height of the tree is \( 80 \) inches.